A Web-Based Educational Simulation Package for Glucose-Insulin Levels in the Human Body

Warning:
"The simulator does not differentiate between people regarding their sex, age, race, or BMI (body mass index); instead it represents an average person. Also, GlucoSim does not take into account intra- and inter-personal variations and it should not be used for making medical decisions."The GlucoSim should only be used for educational purposes.

Model Equations

Overall Model:

Modeling the glucose-insulin interaction requires an understanding of the physiological and metabolic processes that determine the observable behavior. Chemical reactions and transport processes form an integrated network when modeling the glucose-insulin interaction in human body. A number of mathematical models of the insulin-dependent (type-I) diabetes mellitus have been previously reported in the literature (Puckett, 1992; Cobelli 1983; Bergman, 1973; Leaning 1991). We have extended and utilized two mathematical models (Puckett, 1992) based on pharmacokinetic diagrams of glucose and insulin (Figures 1-2), which represent the transport of glucose and insulin through the major vessels to the capillaries.

The glucose diagram (Figure 1) contains tissues including heart, brain, liver, kidney and muscle where the glucose is used for energy. Glucose excretion by kidney and gastrointestinal tract where the exogenous glucose enters the blood, are also included. The diagram for insulin (Figure 2) includes subcutaneous tissue as a source for insulin. It is assumed that pancreas completely lacking the insulin production. Removal and degradation of insulin occurs mostly in liver, kidney and peripheral tissue, they degrade one-half, one-third and one-sixth, respectively, of the insulin presented to them, regardless of the plasma concentration of insulin.

Figure 1. Pharmacokinetic Diagram of the Glucose Model

Figure 2. Pharmacokinetic Diagram of the Insulin Model

Changes in blood flow would change these fractions, but the model flows are constant (Hillman, 1976). Mass balances for the glucose model, which is coupled to the insulin model, result in a set of simultaneous ordinary differential equations.

The model to be presented here is a flow-limited model for diabetes mellitus based on the work of Puckett (1992). A mass balance equation is written for each compartment in the model. The compartments here represent actual body regions. The advantage of this type of modeling is that the model design is based on an understanding of the physiology and simulations that can yield insight into the physiological processes (Hillman, 1976). The main disadvantage of these models is that the personal variations in physiological parameters are not taken into account. Therefore, the outputs are average values. For a typical organ (e.g. liver), mass balance is written as follows:

When the perfusion time scales of the various tissues are considered, they are found to be very small. Therefore, it is assumed that within the individual tissues, changes in the blood glucose and insulin concentrations are fast. The balances for each tissue are in quasi-steady state (i.e. dG/dt @ 0) shortly after a perturbation. Setting dG/dt = 0 in mass balance equations:

(3)

Similarly, setting dI/dt = 0 in the insulin equations:

(4)

Detailed Model:

The quasi-steady state assumption is removed and resulting ordinary differential equations (see bellow) for the mass balances of glucose and insulin are solved simultaneously.

Glucose:

Circulating Blood:

(5)

Kidney:

(6)

Nervous System:

(7)

Periphery:

(8)

Pancreas
& Spleen:

(9)

Gastro-
intestinalTract:

(10)

Heart:

(11)

Liver:

(12)

Insulin:

Circulating Blood:

(13)

Kidney:

(14)

Subcutaneous Tissue:

(15)

Periphery:

(16)

Pancreas
& Spleen:

(17)

Gastro-
intestinal Tract:

(18)

Liver:

(19)

Representation of the Functions for Blood Glucose

Major tissues, including liver, muscle, adipose, heart, kidney and brain, remove glucose from the blood and store or oxidize it for energy. To formulate tissue glucose uptake models, the tissues have been grouped into three categories: (1) Insulin and glucose independent uptake: Central nervous system and red blood cells, (2) Glucose dependent uptake: Kidneys, (3) Insulin and glucose dependent uptake. Remaining major tissues:

Glucose and Insulin Independent Uptake

The glucose uptake rate by the glucose and insulin independent tissues is constant. It will be represented as:

(20)

The brain is the major consumer of this category and is almost totally dependent on glucose as an energy source. The brain contains only 0.1 weight percent glycogen (Fenn and Rahn, 1965) and thus must rely on a minute-to-minute supply of glucose from the blood. Brain glucose uptake rate remains constant with respect to changes in blood glucose concentration except in the case of severe hypoglycemia and is not affected by exercise or by elevated metabolite levels following a meal. The usage by the red blood cells is also constant and relatively small metabolic sink for glucose in the model.

Glucose Dependent Uptake

The only glucose dependent uptake occurs in kidneys where the rate of glucose excretion is equal to its rate of glomerular filtration minus its rate of tubular reabsorption (Robinson, 1967). Urinary excretion of glucose begins when the blood glucose levels exceed a limit of approximately 176 mg/dl. The removal rate of glucose is modeled as:

(21)

where:

GE

glucose excreting rate

GB

circulating plasma glucose concentration (mg/dl)

step function that signifies that glucose excretion does not occur until

Since the glomerular membrane is very permeable and allows everything to pass through except the red blood cells and most proteins, the filtrate concentration of different species such as glucose is approximately that of the plasma (Puckett, 1992). The rate at which glucose is filtered can be calculated by multiplying the plasma glucose concentration by glomerular filtration rate which is 125 ml/min (Guyton, 1976). When the glucose load through the tubules reaches 220 mg/min, the transport mechanisms for the reabsorption process start saturating and some of the glucose leaves in the urine. In the model, this threshold is the factor 1.25 dl/min * 176 mg/dl.

Insulin and Glucose Dependent Uptake

The following model by Puckett(1992) for total glucose uptake (TGU) rate which includes the insulin and glucose dependent uptake and insulin and glucose independent uptake is validated by using the isotope date from Pehling et al. The delay in insulin action in this model has been approximated as a first order system with a time delay.

(22)

(23)

where:

IA

effective insulin concentration or activity

IBD

circulating blood insulin concentration delayed by TD,TGU

TD,TGU

pure time delay for the total glucose uptake

TIA

first order time constant for insulin action activation

k

rate constant for the total glucose uptake

CNU

glucose and insulin independent uptake rate

IB0

initial blood insulin concentration

This model proposed by Puckett (1992) is similar to another model available in the literature (Radziuk et al., 1974), and the parameter estimates show similar results.

Glucose Input Forcing Functions

Glucose enters the blood either through the small intestine of from the liver. Therefore, first carbohydrate ingestion must be supplied to the model as a given input. Secondly, liver glucose production (LGP) needs to be modeled. The model has been adapted from Biermann and Mehnert (1990) to represent glucose absorption rate from the small intestine. This is an empirical representation of the absorption process and instead of using the parameter estimates obtained by Biermann and Mehnert (1990), the date given in Pehling et al. (1984) has been fit to the model by Puckett (1992).

The rate of absorbtion from the small intestine is:

(24)

where:

TA

time constant for absorbtion rate to equilibrate with gastric emptying

F

fraction of meal carbohydrates that actually absorb into the blood

The total amount of glucose in the stomach is:

(25)

where:

TGE

time constant for gastric emptying (min)

The cumulative amount of glucose (i.e. hydrolyzed carbohydrates) from the meal that appears in the stomach is:

(26)

where:

CHOG

rate of hydrolyzed meal carbohydrates that enters the stomach (mg/kg-min)

CHOM

carbohydrate content of the meal (mg/kg)

tM

time of the meal (min)

t

time (min)

Liver Glucose Production

The following submodel has been proposed by Puckett (1992) for the rate of liver glucose production (LGP). The parameters have been obtained by using the experimental data of LGP from Pehling et al. (1984).

The rate of liver glucose production is:

(27)

(28)

where:

GID

GLD x IBD delayed in a first order manner with a time constant kA-1 or kD-1

a1-a4

constants for the quasi-steady state relationships that link the hormone and

the rate limiting steps

GLD

average glucose concentration entering the liver delayed by TD,LGP (mg/dl)

IBD

circulating blood insulin concentration delayed by TD,LGP (mU/dl)

kA-1

time constant for the activation of processes that cause suppression of liver

glucose production (min)

kD-1

time constant for the activation of processes (min)

This model proposed by Puckett (1992) is similar to another model available in the literature (Radziuk et al., 1974), and the parameter estimates show similar results.

Subcutaneous Insulin Transport Models

Puckett and Lightfoot (1995) have determined that a three-pool model similar to the model of Kraegen and Chisholm (1984) is necessary to describe patients data. The following three-pool model has been found to produce an adequate fit to patient data and describes the absorption from an injection of short acting insulin at t = 0 (Puckett, 1992).

(29)

(30)

(31)

where:

IP

total amount of insulin pocket (mU/dl)

IS

insulin concentration in interstitial fluid (mU/dl)

IB

insulin concentration in the capillary blood (mU/dl)

kP

rate constant for the transport of insulin from the pocket into the surrounding

interstitial fluid (min-1)

kS

rate constant for the transport of insulin from the interstitial region to the

capillary blood (min-1)

kB

rate constant for the removal of insulin in the liver and kidney (min-1)

VS,VB

corresponding pool volumes (ml)

,

, and

This model proposed by Puckett (1992) is similar to another model available in the literature (Radziuk et al., 1974), and the parameter estimates show similar results.

Model Development for Healthy People

The only difference from the insulin model of diabetic patients is that injection from subcutaneous tissue is omitted and pancreatic insulin release is included instead. The diagram for the glucose model remains the same. In this work, two pancreatic insulin release (PIR) models have been used for simulating the insulin release by pancreas. The first model is the one developed by Carson and Cramp (1976). In their work, the insulin release has been defined as:

(32)

where:

tBPIR

basal insulin release rate = 4 mU/min

G

glucose concentration

GB

basal glucose concentration = 90 mg/dl

c1

constant (mU/min)(mg/dl)-1

c2

constant (mU/min)(mg/dl)-1

and the superscript 0+ indicates that the arguments in brackets assume a value of zero unless they are greater than zero.

The second model is the islet insulin response model developed by Nomura et al. (1984) for rat islets. When islets are used in place of insulin injections, a model is needed to describe the insulin release rate from an islet and its dependence on plasma glucose levels. This model treats the islet insulin release rate as a proportional-derivative control system. This response is represented by the following general equation:

F. W. Kemmer and M. Berger. Exercise and diabetes mellitus: Physical activity as part of daily life and its role in the treatment of diabetic patients. Interna-tional Journal of Sports Medicine, 4:77-88, 1983.

J. T. Sorensen. A Physiologic Model of Glucose Metabolism in Man and Its Use to Design and Assess Improved Insulin Therapies for Diabetes. PhD thesis, Massachusetts Institute of Technology, Department of Chemical Engineering, 1985.